chain map

Overview

A chain map is a morphism in the category of chain complexes. It is a map that preserves the differential between the abelian groups.

Definition

A chain map $f_{\bullet }:A_{\bullet }\to B_{\bullet }$ is a function between chain complexes such that

1. $f_n:A_n\to B_n$ is a group homomorphism.
2. $f$ is “$d$-linear”. That means the following diagram commutes.

Connection to homology

A chain map $f_{\bullet }:A_{\bullet }\to B_{\bullet }$ induces a homomorphisms of graded abelian groups

$Z_{\bullet }(f):Z_{\bullet }(A)⟶Z_{\bullet }(B)$

such that there is an induced map

$B_{\bullet }(f_{\bullet }):B_{\bullet }(A)⟶B_{\bullet }(B)$

in particular

$H_{\bullet }(f_{\bullet }):H_{\bullet }(A)⟶H_{\bullet }(B)$

In short, a chain map induces a homomorphims between homology groups for a chain complex.